LMFDB - Embedded newform 3344.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3344,2,Mod(1,3344)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3344, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3344.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3344 = 2^{4} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3344.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$26.7019744359$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.744786576.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 $ x^6 - x^5 - 17x^4 + 13x^3 + 69x^2 - 21x - 30
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 836)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.27915$ of defining polynomial
Character	$\chi$	$=$	3344.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.27915 q^{3} +3.53478 q^{5} +2.34459 q^{7} +7.75283 q^{9} +O(q^{10})$	$q-3.27915 q^{3} +3.53478 q^{5} +2.34459 q^{7} +7.75283 q^{9} +1.00000 q^{11} -5.44101 q^{13} -11.5911 q^{15} +3.37635 q^{17} +1.00000 q^{19} -7.68827 q^{21} +5.78459 q^{23} +7.49465 q^{25} -15.5853 q^{27} +9.19463 q^{29} +9.59107 q^{31} -3.27915 q^{33} +8.28761 q^{35} -6.58617 q^{37} +17.8419 q^{39} +2.37145 q^{41} -1.18026 q^{43} +27.4045 q^{45} -1.69320 q^{47} -1.50289 q^{49} -11.0716 q^{51} +2.51125 q^{53} +3.53478 q^{55} -3.27915 q^{57} +4.83925 q^{59} -7.62786 q^{61} +18.1772 q^{63} -19.2328 q^{65} -2.22139 q^{67} -18.9686 q^{69} -11.0379 q^{71} -14.9944 q^{73} -24.5761 q^{75} +2.34459 q^{77} +5.18195 q^{79} +27.8479 q^{81} +4.24982 q^{83} +11.9347 q^{85} -30.1506 q^{87} +6.09139 q^{89} -12.7569 q^{91} -31.4506 q^{93} +3.53478 q^{95} -6.16698 q^{97} +7.75283 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	$ 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) $ 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.27915	−1.89322	−0.946609	−	0.322383i	$-0.895516\pi$
$3$	−3.27915	−1.89322	−0.946609	+	0.322383i	$0.895516\pi$
$4$	0	0
$5$	3.53478	1.58080	0.790400	−	0.612591i	$-0.209873\pi$
$5$	3.53478	1.58080	0.790400	+	0.612591i	$0.209873\pi$
$6$	0	0
$7$	2.34459	0.886172	0.443086	−	0.896479i	$-0.353884\pi$
$7$	2.34459	0.886172	0.443086	+	0.896479i	$0.353884\pi$
$8$	0	0
$9$	7.75283	2.58428
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.44101	−1.50906	−0.754532	−	0.656263i	$-0.772136\pi$
$13$	−5.44101	−1.50906	−0.754532	+	0.656263i	$0.772136\pi$
$14$	0	0
$15$	−11.5911	−2.99280
$16$	0	0
$17$	3.37635	0.818886	0.409443	−	0.912336i	$-0.365723\pi$
$17$	3.37635	0.818886	0.409443	+	0.912336i	$0.365723\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−7.68827	−1.67772
$22$	0	0
$23$	5.78459	1.20617	0.603086	−	0.797676i	$-0.293938\pi$
$23$	5.78459	1.20617	0.603086	+	0.797676i	$0.293938\pi$
$24$	0	0
$25$	7.49465	1.49893
$26$	0	0
$27$	−15.5853	−2.99938
$28$	0	0
$29$	9.19463	1.70740	0.853700	−	0.520766i	$-0.174353\pi$
$29$	9.19463	1.70740	0.853700	+	0.520766i	$0.174353\pi$
$30$	0	0
$31$	9.59107	1.72261	0.861304	−	0.508091i	$-0.169649\pi$
$31$	9.59107	1.72261	0.861304	+	0.508091i	$0.169649\pi$
$32$	0	0
$33$	−3.27915	−0.570827
$34$	0	0
$35$	8.28761	1.40086
$36$	0	0
$37$	−6.58617	−1.08276	−0.541380	−	0.840778i	$-0.682098\pi$
$37$	−6.58617	−1.08276	−0.541380	+	0.840778i	$0.682098\pi$
$38$	0	0
$39$	17.8419	2.85699
$40$	0	0
$41$	2.37145	0.370359	0.185179	−	0.982705i	$-0.440713\pi$
$41$	2.37145	0.370359	0.185179	+	0.982705i	$0.440713\pi$
$42$	0	0
$43$	−1.18026	−0.179988	−0.0899941	−	0.995942i	$-0.528685\pi$
$43$	−1.18026	−0.179988	−0.0899941	+	0.995942i	$0.528685\pi$
$44$	0	0
$45$	27.4045	4.08523
$46$	0	0
$47$	−1.69320	−0.246979	−0.123490	−	0.992346i	$-0.539409\pi$
$47$	−1.69320	−0.246979	−0.123490	+	0.992346i	$0.539409\pi$
$48$	0	0
$49$	−1.50289	−0.214698
$50$	0	0
$51$	−11.0716	−1.55033
$52$	0	0
$53$	2.51125	0.344947	0.172474	−	0.985014i	$-0.444824\pi$
$53$	2.51125	0.344947	0.172474	+	0.985014i	$0.444824\pi$
$54$	0	0
$55$	3.53478	0.476629
$56$	0	0
$57$	−3.27915	−0.434334
$58$	0	0
$59$	4.83925	0.630016	0.315008	−	0.949089i	$-0.397993\pi$
$59$	4.83925	0.630016	0.315008	+	0.949089i	$0.397993\pi$
$60$	0	0
$61$	−7.62786	−0.976647	−0.488324	−	0.872663i	$-0.662391\pi$
$61$	−7.62786	−0.976647	−0.488324	+	0.872663i	$0.662391\pi$
$62$	0	0
$63$	18.1772	2.29012
$64$	0	0
$65$	−19.2328	−2.38553
$66$	0	0
$67$	−2.22139	−0.271386	−0.135693	−	0.990751i	$-0.543326\pi$
$67$	−2.22139	−0.271386	−0.135693	+	0.990751i	$0.543326\pi$
$68$	0	0
$69$	−18.9686	−2.28355
$70$	0	0
$71$	−11.0379	−1.30996	−0.654978	−	0.755648i	$-0.727322\pi$
$71$	−11.0379	−1.30996	−0.654978	+	0.755648i	$0.727322\pi$
$72$	0	0
$73$	−14.9944	−1.75496	−0.877482	−	0.479610i	$-0.840778\pi$
$73$	−14.9944	−1.75496	−0.877482	+	0.479610i	$0.840778\pi$
$74$	0	0
$75$	−24.5761	−2.83780
$76$	0	0
$77$	2.34459	0.267191
$78$	0	0
$79$	5.18195	0.583015	0.291507	−	0.956569i	$-0.405843\pi$
$79$	5.18195	0.583015	0.291507	+	0.956569i	$0.405843\pi$
$80$	0	0
$81$	27.8479	3.09421
$82$	0	0
$83$	4.24982	0.466478	0.233239	−	0.972419i	$-0.425068\pi$
$83$	4.24982	0.466478	0.233239	+	0.972419i	$0.425068\pi$
$84$	0	0
$85$	11.9347	1.29450
$86$	0	0
$87$	−30.1506	−3.23248
$88$	0	0
$89$	6.09139	0.645686	0.322843	−	0.946453i	$-0.395361\pi$
$89$	6.09139	0.645686	0.322843	+	0.946453i	$0.395361\pi$
$90$	0	0
$91$	−12.7569	−1.33729
$92$	0	0
$93$	−31.4506	−3.26127
$94$	0	0
$95$	3.53478	0.362661
$96$	0	0
$97$	−6.16698	−0.626162	−0.313081	−	0.949726i	$-0.601361\pi$
$97$	−6.16698	−0.626162	−0.313081	+	0.949726i	$0.601361\pi$
$98$	0	0
$99$	7.75283	0.779189
$100$	0	0
$101$	6.86510	0.683103	0.341551	−	0.939863i	$-0.389048\pi$
$101$	6.86510	0.683103	0.341551	+	0.939863i	$0.389048\pi$
$102$	0	0
$103$	−12.8156	−1.26276	−0.631380	−	0.775473i	$-0.717511\pi$
$103$	−12.8156	−1.26276	−0.631380	+	0.775473i	$0.717511\pi$
$104$	0	0
$105$	−27.1763	−2.65214
$106$	0	0
$107$	−3.49076	−0.337465	−0.168732	−	0.985662i	$-0.553967\pi$
$107$	−3.49076	−0.337465	−0.168732	+	0.985662i	$0.553967\pi$
$108$	0	0
$109$	19.9904	1.91473	0.957366	−	0.288877i	$-0.0932817\pi$
$109$	19.9904	1.91473	0.957366	+	0.288877i	$0.0932817\pi$
$110$	0	0
$111$	21.5971	2.04990
$112$	0	0
$113$	14.0212	1.31900	0.659502	−	0.751703i	$-0.270767\pi$
$113$	14.0212	1.31900	0.659502	+	0.751703i	$0.270767\pi$
$114$	0	0
$115$	20.4473	1.90672
$116$	0	0
$117$	−42.1832	−3.89984
$118$	0	0
$119$	7.91617	0.725674
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.77636	−0.701171
$124$	0	0
$125$	8.81804	0.788710
$126$	0	0
$127$	5.54742	0.492254	0.246127	−	0.969238i	$-0.420842\pi$
$127$	5.54742	0.492254	0.246127	+	0.969238i	$0.420842\pi$
$128$	0	0
$129$	3.87026	0.340757
$130$	0	0
$131$	11.6672	1.01937	0.509685	−	0.860361i	$-0.329762\pi$
$131$	11.6672	1.01937	0.509685	+	0.860361i	$0.329762\pi$
$132$	0	0
$133$	2.34459	0.203302
$134$	0	0
$135$	−55.0904	−4.74143
$136$	0	0
$137$	−2.82797	−0.241610	−0.120805	−	0.992676i	$-0.538548\pi$
$137$	−2.82797	−0.241610	−0.120805	+	0.992676i	$0.538548\pi$
$138$	0	0
$139$	−15.4993	−1.31464	−0.657318	−	0.753614i	$-0.728309\pi$
$139$	−15.4993	−1.31464	−0.657318	+	0.753614i	$0.728309\pi$
$140$	0	0
$141$	5.55227	0.467585
$142$	0	0
$143$	−5.44101	−0.455000
$144$	0	0
$145$	32.5010	2.69906
$146$	0	0
$147$	4.92820	0.406471
$148$	0	0
$149$	−0.234586	−0.0192180	−0.00960900	−	0.999954i	$-0.503059\pi$
$149$	−0.234586	−0.0192180	−0.00960900	+	0.999954i	$0.503059\pi$
$150$	0	0
$151$	−17.3110	−1.40875	−0.704375	−	0.709828i	$-0.748773\pi$
$151$	−17.3110	−1.40875	−0.704375	+	0.709828i	$0.748773\pi$
$152$	0	0
$153$	26.1763	2.11623
$154$	0	0
$155$	33.9023	2.72310
$156$	0	0
$157$	−10.8360	−0.864806	−0.432403	−	0.901680i	$-0.642334\pi$
$157$	−10.8360	−0.864806	−0.432403	+	0.901680i	$0.642334\pi$
$158$	0	0
$159$	−8.23478	−0.653060
$160$	0	0
$161$	13.5625	1.06888
$162$	0	0
$163$	8.95716	0.701579	0.350789	−	0.936454i	$-0.385913\pi$
$163$	8.95716	0.701579	0.350789	+	0.936454i	$0.385913\pi$
$164$	0	0
$165$	−11.5911	−0.902364
$166$	0	0
$167$	14.1193	1.09258	0.546291	−	0.837596i	$-0.316039\pi$
$167$	14.1193	1.09258	0.546291	+	0.837596i	$0.316039\pi$
$168$	0	0
$169$	16.6046	1.27728
$170$	0	0
$171$	7.75283	0.592874
$172$	0	0
$173$	2.56411	0.194946	0.0974729	−	0.995238i	$-0.468924\pi$
$173$	2.56411	0.194946	0.0974729	+	0.995238i	$0.468924\pi$
$174$	0	0
$175$	17.5719	1.32831
$176$	0	0
$177$	−15.8686	−1.19276
$178$	0	0
$179$	−0.455979	−0.0340815	−0.0170407	−	0.999855i	$-0.505424\pi$
$179$	−0.455979	−0.0340815	−0.0170407	+	0.999855i	$0.505424\pi$
$180$	0	0
$181$	−8.41804	−0.625708	−0.312854	−	0.949801i	$-0.601285\pi$
$181$	−8.41804	−0.625708	−0.312854	+	0.949801i	$0.601285\pi$
$182$	0	0
$183$	25.0129	1.84901
$184$	0	0
$185$	−23.2807	−1.71163
$186$	0	0
$187$	3.37635	0.246903
$188$	0	0
$189$	−36.5411	−2.65797
$190$	0	0
$191$	−17.9694	−1.30022	−0.650110	−	0.759840i	$-0.725277\pi$
$191$	−17.9694	−1.30022	−0.650110	+	0.759840i	$0.725277\pi$
$192$	0	0
$193$	17.2504	1.24171	0.620854	−	0.783926i	$-0.286786\pi$
$193$	17.2504	1.24171	0.620854	+	0.783926i	$0.286786\pi$
$194$	0	0
$195$	63.0671	4.51633
$196$	0	0
$197$	−6.88359	−0.490435	−0.245218	−	0.969468i	$-0.578859\pi$
$197$	−6.88359	−0.490435	−0.245218	+	0.969468i	$0.578859\pi$
$198$	0	0
$199$	15.0531	1.06708	0.533542	−	0.845773i	$-0.320860\pi$
$199$	15.0531	1.06708	0.533542	+	0.845773i	$0.320860\pi$
$200$	0	0
$201$	7.28429	0.513794
$202$	0	0
$203$	21.5576	1.51305
$204$	0	0
$205$	8.38256	0.585464
$206$	0	0
$207$	44.8470	3.11708
$208$	0	0
$209$	1.00000	0.0691714
$210$	0	0
$211$	−19.2475	−1.32505	−0.662526	−	0.749039i	$-0.730516\pi$
$211$	−19.2475	−1.32505	−0.662526	+	0.749039i	$0.730516\pi$
$212$	0	0
$213$	36.1949	2.48003
$214$	0	0
$215$	−4.17196	−0.284525
$216$	0	0
$217$	22.4871	1.52653
$218$	0	0
$219$	49.1690	3.32253
$220$	0	0
$221$	−18.3708	−1.23575
$222$	0	0
$223$	15.6477	1.04785	0.523923	−	0.851765i	$-0.324468\pi$
$223$	15.6477	1.04785	0.523923	+	0.851765i	$0.324468\pi$
$224$	0	0
$225$	58.1048	3.87365
$226$	0	0
$227$	25.9837	1.72460	0.862300	−	0.506398i	$-0.169023\pi$
$227$	25.9837	1.72460	0.862300	+	0.506398i	$0.169023\pi$
$228$	0	0
$229$	−19.9887	−1.32089	−0.660445	−	0.750874i	$-0.729632\pi$
$229$	−19.9887	−1.32089	−0.660445	+	0.750874i	$0.729632\pi$
$230$	0	0
$231$	−7.68827	−0.505851
$232$	0	0
$233$	16.7578	1.09784	0.548920	−	0.835875i	$-0.315039\pi$
$233$	16.7578	1.09784	0.548920	+	0.835875i	$0.315039\pi$
$234$	0	0
$235$	−5.98510	−0.390425
$236$	0	0
$237$	−16.9924	−1.10377
$238$	0	0
$239$	6.69553	0.433098	0.216549	−	0.976272i	$-0.430520\pi$
$239$	6.69553	0.433098	0.216549	+	0.976272i	$0.430520\pi$
$240$	0	0
$241$	9.95052	0.640969	0.320485	−	0.947254i	$-0.396154\pi$
$241$	9.95052	0.640969	0.320485	+	0.947254i	$0.396154\pi$
$242$	0	0
$243$	−44.5618	−2.85864
$244$	0	0
$245$	−5.31238	−0.339395
$246$	0	0
$247$	−5.44101	−0.346203
$248$	0	0
$249$	−13.9358	−0.883145
$250$	0	0
$251$	−2.64146	−0.166728	−0.0833639	−	0.996519i	$-0.526566\pi$
$251$	−2.64146	−0.166728	−0.0833639	+	0.996519i	$0.526566\pi$
$252$	0	0
$253$	5.78459	0.363674
$254$	0	0
$255$	−39.1355	−2.45076
$256$	0	0
$257$	−2.24328	−0.139932	−0.0699659	−	0.997549i	$-0.522289\pi$
$257$	−2.24328	−0.139932	−0.0699659	+	0.997549i	$0.522289\pi$
$258$	0	0
$259$	−15.4419	−0.959512
$260$	0	0
$261$	71.2844	4.41239
$262$	0	0
$263$	8.31828	0.512927	0.256464	−	0.966554i	$-0.417443\pi$
$263$	8.31828	0.512927	0.256464	+	0.966554i	$0.417443\pi$
$264$	0	0
$265$	8.87672	0.545293
$266$	0	0
$267$	−19.9746	−1.22243
$268$	0	0
$269$	−31.0749	−1.89467	−0.947334	−	0.320248i	$-0.896234\pi$
$269$	−31.0749	−1.89467	−0.947334	+	0.320248i	$0.896234\pi$
$270$	0	0
$271$	23.9002	1.45184	0.725918	−	0.687781i	$-0.241415\pi$
$271$	23.9002	1.45184	0.725918	+	0.687781i	$0.241415\pi$
$272$	0	0
$273$	41.8320	2.53179
$274$	0	0
$275$	7.49465	0.451945
$276$	0	0
$277$	−1.66668	−0.100141	−0.0500705	−	0.998746i	$-0.515945\pi$
$277$	−1.66668	−0.100141	−0.0500705	+	0.998746i	$0.515945\pi$
$278$	0	0
$279$	74.3580	4.45170
$280$	0	0
$281$	13.3003	0.793432	0.396716	−	0.917941i	$-0.370150\pi$
$281$	13.3003	0.793432	0.396716	+	0.917941i	$0.370150\pi$
$282$	0	0
$283$	−13.2362	−0.786811	−0.393406	−	0.919365i	$-0.628703\pi$
$283$	−13.2362	−0.786811	−0.393406	+	0.919365i	$0.628703\pi$
$284$	0	0
$285$	−11.5911	−0.686596
$286$	0	0
$287$	5.56009	0.328202
$288$	0	0
$289$	−5.60024	−0.329426
$290$	0	0
$291$	20.2225	1.18546
$292$	0	0
$293$	−8.08113	−0.472105	−0.236052	−	0.971740i	$-0.575854\pi$
$293$	−8.08113	−0.472105	−0.236052	+	0.971740i	$0.575854\pi$
$294$	0	0
$295$	17.1057	0.995930
$296$	0	0
$297$	−15.5853	−0.904349
$298$	0	0
$299$	−31.4740	−1.82019
$300$	0	0
$301$	−2.76723	−0.159501
$302$	0	0
$303$	−22.5117	−1.29326
$304$	0	0
$305$	−26.9628	−1.54388
$306$	0	0
$307$	4.12730	0.235557	0.117779	−	0.993040i	$-0.462423\pi$
$307$	4.12730	0.235557	0.117779	+	0.993040i	$0.462423\pi$
$308$	0	0
$309$	42.0244	2.39068
$310$	0	0
$311$	−10.7342	−0.608682	−0.304341	−	0.952563i	$-0.598436\pi$
$311$	−10.7342	−0.608682	−0.304341	+	0.952563i	$0.598436\pi$
$312$	0	0
$313$	7.22131	0.408172	0.204086	−	0.978953i	$-0.434578\pi$
$313$	7.22131	0.408172	0.204086	+	0.978953i	$0.434578\pi$
$314$	0	0
$315$	64.2525	3.62022
$316$	0	0
$317$	17.7601	0.997506	0.498753	−	0.866744i	$-0.333792\pi$
$317$	17.7601	0.997506	0.498753	+	0.866744i	$0.333792\pi$
$318$	0	0
$319$	9.19463	0.514800
$320$	0	0
$321$	11.4467	0.638895
$322$	0	0
$323$	3.37635	0.187865
$324$	0	0
$325$	−40.7785	−2.26198
$326$	0	0
$327$	−65.5515	−3.62501
$328$	0	0
$329$	−3.96987	−0.218866
$330$	0	0
$331$	−5.86554	−0.322399	−0.161200	−	0.986922i	$-0.551536\pi$
$331$	−5.86554	−0.322399	−0.161200	+	0.986922i	$0.551536\pi$
$332$	0	0
$333$	−51.0615	−2.79815
$334$	0	0
$335$	−7.85213	−0.429008
$336$	0	0
$337$	−6.27598	−0.341874	−0.170937	−	0.985282i	$-0.554680\pi$
$337$	−6.27598	−0.341874	−0.170937	+	0.985282i	$0.554680\pi$
$338$	0	0
$339$	−45.9776	−2.49716
$340$	0	0
$341$	9.59107	0.519386
$342$	0	0
$343$	−19.9358	−1.07643
$344$	0	0
$345$	−67.0496	−3.60983
$346$	0	0
$347$	12.8165	0.688025	0.344012	−	0.938965i	$-0.388214\pi$
$347$	12.8165	0.688025	0.344012	+	0.938965i	$0.388214\pi$
$348$	0	0
$349$	8.94535	0.478834	0.239417	−	0.970917i	$-0.423044\pi$
$349$	8.94535	0.478834	0.239417	+	0.970917i	$0.423044\pi$
$350$	0	0
$351$	84.7996	4.52627
$352$	0	0
$353$	33.8419	1.80122	0.900611	−	0.434626i	$-0.143119\pi$
$353$	33.8419	1.80122	0.900611	+	0.434626i	$0.143119\pi$
$354$	0	0
$355$	−39.0165	−2.07078
$356$	0	0
$357$	−25.9583	−1.37386
$358$	0	0
$359$	2.70804	0.142925	0.0714625	−	0.997443i	$-0.477233\pi$
$359$	2.70804	0.142925	0.0714625	+	0.997443i	$0.477233\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−3.27915	−0.172111
$364$	0	0
$365$	−53.0019	−2.77425
$366$	0	0
$367$	−22.5295	−1.17603	−0.588016	−	0.808850i	$-0.700091\pi$
$367$	−22.5295	−1.17603	−0.588016	+	0.808850i	$0.700091\pi$
$368$	0	0
$369$	18.3855	0.957111
$370$	0	0
$371$	5.88786	0.305683
$372$	0	0
$373$	25.5197	1.32136	0.660680	−	0.750668i	$-0.270268\pi$
$373$	25.5197	1.32136	0.660680	+	0.750668i	$0.270268\pi$
$374$	0	0
$375$	−28.9157	−1.49320
$376$	0	0
$377$	−50.0281	−2.57658
$378$	0	0
$379$	12.0860	0.620818	0.310409	−	0.950603i	$-0.399534\pi$
$379$	12.0860	0.620818	0.310409	+	0.950603i	$0.399534\pi$
$380$	0	0
$381$	−18.1908	−0.931944
$382$	0	0
$383$	−19.4116	−0.991885	−0.495943	−	0.868355i	$-0.665177\pi$
$383$	−19.4116	−0.991885	−0.495943	+	0.868355i	$0.665177\pi$
$384$	0	0
$385$	8.28761	0.422376
$386$	0	0
$387$	−9.15037	−0.465139
$388$	0	0
$389$	11.2668	0.571250	0.285625	−	0.958342i	$-0.407799\pi$
$389$	11.2668	0.571250	0.285625	+	0.958342i	$0.407799\pi$
$390$	0	0
$391$	19.5308	0.987716
$392$	0	0
$393$	−38.2586	−1.92989
$394$	0	0
$395$	18.3170	0.921630
$396$	0	0
$397$	11.9391	0.599204	0.299602	−	0.954064i	$-0.403146\pi$
$397$	11.9391	0.599204	0.299602	+	0.954064i	$0.403146\pi$
$398$	0	0
$399$	−7.68827	−0.384895
$400$	0	0
$401$	9.90453	0.494608	0.247304	−	0.968938i	$-0.420455\pi$
$401$	9.90453	0.494608	0.247304	+	0.968938i	$0.420455\pi$
$402$	0	0
$403$	−52.1851	−2.59953
$404$	0	0
$405$	98.4362	4.89134
$406$	0	0
$407$	−6.58617	−0.326464
$408$	0	0
$409$	11.1425	0.550960	0.275480	−	0.961307i	$-0.411163\pi$
$409$	11.1425	0.550960	0.275480	+	0.961307i	$0.411163\pi$
$410$	0	0
$411$	9.27336	0.457421
$412$	0	0
$413$	11.3461	0.558303
$414$	0	0
$415$	15.0222	0.737409
$416$	0	0
$417$	50.8246	2.48889
$418$	0	0
$419$	7.88093	0.385009	0.192504	−	0.981296i	$-0.438339\pi$
$419$	7.88093	0.385009	0.192504	+	0.981296i	$0.438339\pi$
$420$	0	0
$421$	37.8040	1.84245	0.921226	−	0.389027i	$-0.127189\pi$
$421$	37.8040	1.84245	0.921226	+	0.389027i	$0.127189\pi$
$422$	0	0
$423$	−13.1271	−0.638262
$424$	0	0
$425$	25.3046	1.22745
$426$	0	0
$427$	−17.8842	−0.865478
$428$	0	0
$429$	17.8419	0.861415
$430$	0	0
$431$	9.80267	0.472178	0.236089	−	0.971731i	$-0.424134\pi$
$431$	9.80267	0.472178	0.236089	+	0.971731i	$0.424134\pi$
$432$	0	0
$433$	−22.1855	−1.06617	−0.533083	−	0.846063i	$-0.678967\pi$
$433$	−22.1855	−1.06617	−0.533083	+	0.846063i	$0.678967\pi$
$434$	0	0
$435$	−106.576	−5.10991
$436$	0	0
$437$	5.78459	0.276715
$438$	0	0
$439$	−18.9073	−0.902396	−0.451198	−	0.892424i	$-0.649003\pi$
$439$	−18.9073	−0.902396	−0.451198	+	0.892424i	$0.649003\pi$
$440$	0	0
$441$	−11.6516	−0.554840
$442$	0	0
$443$	−4.46822	−0.212292	−0.106146	−	0.994351i	$-0.533851\pi$
$443$	−4.46822	−0.212292	−0.106146	+	0.994351i	$0.533851\pi$
$444$	0	0
$445$	21.5317	1.02070
$446$	0	0
$447$	0.769242	0.0363839
$448$	0	0
$449$	−28.6131	−1.35033	−0.675167	−	0.737665i	$-0.735929\pi$
$449$	−28.6131	−1.35033	−0.675167	+	0.737665i	$0.735929\pi$
$450$	0	0
$451$	2.37145	0.111667
$452$	0	0
$453$	56.7654	2.66707
$454$	0	0
$455$	−45.0930	−2.11399
$456$	0	0
$457$	−14.8196	−0.693232	−0.346616	−	0.938007i	$-0.612669\pi$
$457$	−14.8196	−0.693232	−0.346616	+	0.938007i	$0.612669\pi$
$458$	0	0
$459$	−52.6213	−2.45615
$460$	0	0
$461$	36.4495	1.69762	0.848811	−	0.528697i	$-0.177319\pi$
$461$	36.4495	1.69762	0.848811	+	0.528697i	$0.177319\pi$
$462$	0	0
$463$	32.9591	1.53174	0.765871	−	0.642995i	$-0.222308\pi$
$463$	32.9591	1.53174	0.765871	+	0.642995i	$0.222308\pi$
$464$	0	0
$465$	−111.171	−5.15542
$466$	0	0
$467$	−6.28650	−0.290905	−0.145452	−	0.989365i	$-0.546464\pi$
$467$	−6.28650	−0.290905	−0.145452	+	0.989365i	$0.546464\pi$
$468$	0	0
$469$	−5.20826	−0.240495
$470$	0	0
$471$	35.5329	1.63727
$472$	0	0
$473$	−1.18026	−0.0542685
$474$	0	0
$475$	7.49465	0.343878
$476$	0	0
$477$	19.4693	0.891439
$478$	0	0
$479$	18.8832	0.862797	0.431399	−	0.902161i	$-0.358020\pi$
$479$	18.8832	0.862797	0.431399	+	0.902161i	$0.358020\pi$
$480$	0	0
$481$	35.8354	1.63396
$482$	0	0
$483$	−44.4735	−2.02362
$484$	0	0
$485$	−21.7989	−0.989837
$486$	0	0
$487$	−24.5268	−1.11142	−0.555708	−	0.831377i	$-0.687553\pi$
$487$	−24.5268	−1.11142	−0.555708	+	0.831377i	$0.687553\pi$
$488$	0	0
$489$	−29.3719	−1.32824
$490$	0	0
$491$	34.8056	1.57075	0.785377	−	0.619017i	$-0.212469\pi$
$491$	34.8056	1.57075	0.785377	+	0.619017i	$0.212469\pi$
$492$	0	0
$493$	31.0443	1.39816
$494$	0	0
$495$	27.4045	1.23174
$496$	0	0
$497$	−25.8794	−1.16085
$498$	0	0
$499$	9.63700	0.431411	0.215706	−	0.976458i	$-0.430795\pi$
$499$	9.63700	0.431411	0.215706	+	0.976458i	$0.430795\pi$
$500$	0	0
$501$	−46.2992	−2.06850
$502$	0	0
$503$	−32.1848	−1.43505	−0.717524	−	0.696534i	$-0.754724\pi$
$503$	−32.1848	−1.43505	−0.717524	+	0.696534i	$0.754724\pi$
$504$	0	0
$505$	24.2666	1.07985
$506$	0	0
$507$	−54.4490	−2.41816
$508$	0	0
$509$	13.3645	0.592371	0.296185	−	0.955131i	$-0.404285\pi$
$509$	13.3645	0.592371	0.296185	+	0.955131i	$0.404285\pi$
$510$	0	0
$511$	−35.1558	−1.55520
$512$	0	0
$513$	−15.5853	−0.688106
$514$	0	0
$515$	−45.3004	−1.99617
$516$	0	0
$517$	−1.69320	−0.0744670
$518$	0	0
$519$	−8.40811	−0.369075
$520$	0	0
$521$	15.8705	0.695298	0.347649	−	0.937625i	$-0.386980\pi$
$521$	15.8705	0.695298	0.347649	+	0.937625i	$0.386980\pi$
$522$	0	0
$523$	17.2073	0.752423	0.376211	−	0.926534i	$-0.377227\pi$
$523$	17.2073	0.752423	0.376211	+	0.926534i	$0.377227\pi$
$524$	0	0
$525$	−57.6209	−2.51478
$526$	0	0
$527$	32.3828	1.41062
$528$	0	0
$529$	10.4615	0.454849
$530$	0	0
$531$	37.5179	1.62814
$532$	0	0
$533$	−12.9031	−0.558896
$534$	0	0
$535$	−12.3391	−0.533465
$536$	0	0
$537$	1.49522	0.0645237
$538$	0	0
$539$	−1.50289	−0.0647340
$540$	0	0
$541$	35.0905	1.50866	0.754330	−	0.656496i	$-0.227962\pi$
$541$	35.0905	1.50866	0.754330	+	0.656496i	$0.227962\pi$
$542$	0	0
$543$	27.6040	1.18460
$544$	0	0
$545$	70.6616	3.02681
$546$	0	0
$547$	2.78485	0.119072	0.0595358	−	0.998226i	$-0.481038\pi$
$547$	2.78485	0.119072	0.0595358	+	0.998226i	$0.481038\pi$
$548$	0	0
$549$	−59.1375	−2.52393
$550$	0	0
$551$	9.19463	0.391704
$552$	0	0
$553$	12.1496	0.516652
$554$	0	0
$555$	76.3408	3.24049
$556$	0	0
$557$	2.49963	0.105913	0.0529564	−	0.998597i	$-0.483136\pi$
$557$	2.49963	0.105913	0.0529564	+	0.998597i	$0.483136\pi$
$558$	0	0
$559$	6.42181	0.271614
$560$	0	0
$561$	−11.0716	−0.467442
$562$	0	0
$563$	−6.06488	−0.255604	−0.127802	−	0.991800i	$-0.540792\pi$
$563$	−6.06488	−0.255604	−0.127802	+	0.991800i	$0.540792\pi$
$564$	0	0
$565$	49.5618	2.08508
$566$	0	0
$567$	65.2920	2.74201
$568$	0	0
$569$	−14.9447	−0.626513	−0.313256	−	0.949669i	$-0.601420\pi$
$569$	−14.9447	−0.626513	−0.313256	+	0.949669i	$0.601420\pi$
$570$	0	0
$571$	−27.3521	−1.14465	−0.572325	−	0.820027i	$-0.693958\pi$
$571$	−27.3521	−1.14465	−0.572325	+	0.820027i	$0.693958\pi$
$572$	0	0
$573$	58.9244	2.46160
$574$	0	0
$575$	43.3535	1.80797
$576$	0	0
$577$	−1.83560	−0.0764170	−0.0382085	−	0.999270i	$-0.512165\pi$
$577$	−1.83560	−0.0764170	−0.0382085	+	0.999270i	$0.512165\pi$
$578$	0	0
$579$	−56.5666	−2.35083
$580$	0	0
$581$	9.96409	0.413380
$582$	0	0
$583$	2.51125	0.104005
$584$	0	0
$585$	−149.108	−6.16487
$586$	0	0
$587$	−31.1835	−1.28708	−0.643540	−	0.765412i	$-0.722535\pi$
$587$	−31.1835	−1.28708	−0.643540	+	0.765412i	$0.722535\pi$
$588$	0	0
$589$	9.59107	0.395193
$590$	0	0
$591$	22.5723	0.928501
$592$	0	0
$593$	20.4998	0.841827	0.420913	−	0.907101i	$-0.361710\pi$
$593$	20.4998	0.841827	0.420913	+	0.907101i	$0.361710\pi$
$594$	0	0
$595$	27.9819	1.14715
$596$	0	0
$597$	−49.3613	−2.02023
$598$	0	0
$599$	21.8573	0.893065	0.446532	−	0.894768i	$-0.352659\pi$
$599$	21.8573	0.893065	0.446532	+	0.894768i	$0.352659\pi$
$600$	0	0
$601$	2.12818	0.0868102	0.0434051	−	0.999058i	$-0.486179\pi$
$601$	2.12818	0.0868102	0.0434051	+	0.999058i	$0.486179\pi$
$602$	0	0
$603$	−17.2221	−0.701338
$604$	0	0
$605$	3.53478	0.143709
$606$	0	0
$607$	11.3933	0.462439	0.231219	−	0.972902i	$-0.425729\pi$
$607$	11.3933	0.462439	0.231219	+	0.972902i	$0.425729\pi$
$608$	0	0
$609$	−70.6908	−2.86454
$610$	0	0
$611$	9.21273	0.372707
$612$	0	0
$613$	−44.3930	−1.79302	−0.896508	−	0.443028i	$-0.853904\pi$
$613$	−44.3930	−1.79302	−0.896508	+	0.443028i	$0.853904\pi$
$614$	0	0
$615$	−27.4877	−1.10841
$616$	0	0
$617$	37.3951	1.50547	0.752736	−	0.658323i	$-0.228734\pi$
$617$	37.3951	1.50547	0.752736	+	0.658323i	$0.228734\pi$
$618$	0	0
$619$	15.6910	0.630673	0.315337	−	0.948980i	$-0.397883\pi$
$619$	15.6910	0.630673	0.315337	+	0.948980i	$0.397883\pi$
$620$	0	0
$621$	−90.1544	−3.61777
$622$	0	0
$623$	14.2818	0.572189
$624$	0	0
$625$	−6.30344	−0.252138
$626$	0	0
$627$	−3.27915	−0.130957
$628$	0	0
$629$	−22.2372	−0.886657
$630$	0	0
$631$	7.72618	0.307574	0.153787	−	0.988104i	$-0.450853\pi$
$631$	7.72618	0.307574	0.153787	+	0.988104i	$0.450853\pi$
$632$	0	0
$633$	63.1154	2.50861
$634$	0	0
$635$	19.6089	0.778155
$636$	0	0
$637$	8.17723	0.323994
$638$	0	0
$639$	−85.5749	−3.38529
$640$	0	0
$641$	−7.26960	−0.287132	−0.143566	−	0.989641i	$-0.545857\pi$
$641$	−7.26960	−0.287132	−0.143566	+	0.989641i	$0.545857\pi$
$642$	0	0
$643$	6.00629	0.236865	0.118432	−	0.992962i	$-0.462213\pi$
$643$	6.00629	0.236865	0.118432	+	0.992962i	$0.462213\pi$
$644$	0	0
$645$	13.6805	0.538669
$646$	0	0
$647$	−46.0802	−1.81160	−0.905801	−	0.423704i	$-0.860730\pi$
$647$	−46.0802	−1.81160	−0.905801	+	0.423704i	$0.860730\pi$
$648$	0	0
$649$	4.83925	0.189957
$650$	0	0
$651$	−73.7388	−2.89005
$652$	0	0
$653$	−37.2226	−1.45663	−0.728317	−	0.685240i	$-0.759697\pi$
$653$	−37.2226	−1.45663	−0.728317	+	0.685240i	$0.759697\pi$
$654$	0	0
$655$	41.2410	1.61142
$656$	0	0
$657$	−116.249	−4.53531
$658$	0	0
$659$	−34.9204	−1.36030	−0.680152	−	0.733071i	$-0.738086\pi$
$659$	−34.9204	−1.36030	−0.680152	+	0.733071i	$0.738086\pi$
$660$	0	0
$661$	−18.6194	−0.724211	−0.362105	−	0.932137i	$-0.617942\pi$
$661$	−18.6194	−0.724211	−0.362105	+	0.932137i	$0.617942\pi$
$662$	0	0
$663$	60.2405	2.33955
$664$	0	0
$665$	8.28761	0.321380
$666$	0	0
$667$	53.1872	2.05942
$668$	0	0
$669$	−51.3111	−1.98380
$670$	0	0
$671$	−7.62786	−0.294470
$672$	0	0
$673$	−39.6581	−1.52871	−0.764355	−	0.644796i	$-0.776942\pi$
$673$	−39.6581	−1.52871	−0.764355	+	0.644796i	$0.776942\pi$
$674$	0	0
$675$	−116.806	−4.49587
$676$	0	0
$677$	−43.8777	−1.68636	−0.843179	−	0.537632i	$-0.819319\pi$
$677$	−43.8777	−1.68636	−0.843179	+	0.537632i	$0.819319\pi$
$678$	0	0
$679$	−14.4591	−0.554888
$680$	0	0
$681$	−85.2046	−3.26505
$682$	0	0
$683$	5.57257	0.213228	0.106614	−	0.994300i	$-0.465999\pi$
$683$	5.57257	0.213228	0.106614	+	0.994300i	$0.465999\pi$
$684$	0	0
$685$	−9.99626	−0.381937
$686$	0	0
$687$	65.5460	2.50073
$688$	0	0
$689$	−13.6638	−0.520547
$690$	0	0
$691$	3.03739	0.115548	0.0577739	−	0.998330i	$-0.481600\pi$
$691$	3.03739	0.115548	0.0577739	+	0.998330i	$0.481600\pi$
$692$	0	0
$693$	18.1772	0.690496
$694$	0	0
$695$	−54.7866	−2.07818
$696$	0	0
$697$	8.00687	0.303282
$698$	0	0
$699$	−54.9514	−2.07845
$700$	0	0
$701$	6.85374	0.258862	0.129431	−	0.991588i	$-0.458685\pi$
$701$	6.85374	0.258862	0.129431	+	0.991588i	$0.458685\pi$
$702$	0	0
$703$	−6.58617	−0.248402
$704$	0	0
$705$	19.6260	0.739159
$706$	0	0
$707$	16.0959	0.605347
$708$	0	0
$709$	22.4947	0.844804	0.422402	−	0.906409i	$-0.361187\pi$
$709$	22.4947	0.844804	0.422402	+	0.906409i	$0.361187\pi$
$710$	0	0
$711$	40.1748	1.50667
$712$	0	0
$713$	55.4805	2.07776
$714$	0	0
$715$	−19.2328	−0.719264
$716$	0	0
$717$	−21.9557	−0.819949
$718$	0	0
$719$	−16.5930	−0.618816	−0.309408	−	0.950929i	$-0.600131\pi$
$719$	−16.5930	−0.618816	−0.309408	+	0.950929i	$0.600131\pi$
$720$	0	0
$721$	−30.0474	−1.11902
$722$	0	0
$723$	−32.6293	−1.21350
$724$	0	0
$725$	68.9105	2.55927
$726$	0	0
$727$	−12.7104	−0.471402	−0.235701	−	0.971826i	$-0.575739\pi$
$727$	−12.7104	−0.471402	−0.235701	+	0.971826i	$0.575739\pi$
$728$	0	0
$729$	62.5811	2.31782
$730$	0	0
$731$	−3.98498	−0.147390
$732$	0	0
$733$	−19.8202	−0.732077	−0.366039	−	0.930600i	$-0.619286\pi$
$733$	−19.8202	−0.732077	−0.366039	+	0.930600i	$0.619286\pi$
$734$	0	0
$735$	17.4201	0.642550
$736$	0	0
$737$	−2.22139	−0.0818261
$738$	0	0
$739$	48.1166	1.77000	0.885000	−	0.465592i	$-0.154158\pi$
$739$	48.1166	1.77000	0.885000	+	0.465592i	$0.154158\pi$
$740$	0	0
$741$	17.8419	0.655438
$742$	0	0
$743$	−35.5155	−1.30294	−0.651468	−	0.758676i	$-0.725847\pi$
$743$	−35.5155	−1.30294	−0.651468	+	0.758676i	$0.725847\pi$
$744$	0	0
$745$	−0.829208	−0.0303798
$746$	0	0
$747$	32.9481	1.20551
$748$	0	0
$749$	−8.18441	−0.299052
$750$	0	0
$751$	8.43033	0.307627	0.153814	−	0.988100i	$-0.450844\pi$
$751$	8.43033	0.307627	0.153814	+	0.988100i	$0.450844\pi$
$752$	0	0
$753$	8.66176	0.315652
$754$	0	0
$755$	−61.1906	−2.22695
$756$	0	0
$757$	−39.6118	−1.43972	−0.719858	−	0.694121i	$-0.755793\pi$
$757$	−39.6118	−1.43972	−0.719858	+	0.694121i	$0.755793\pi$
$758$	0	0
$759$	−18.9686	−0.688515
$760$	0	0
$761$	−40.8269	−1.47998	−0.739988	−	0.672621i	$-0.765169\pi$
$761$	−40.8269	−1.47998	−0.739988	+	0.672621i	$0.765169\pi$
$762$	0	0
$763$	46.8693	1.69678
$764$	0	0
$765$	92.5274	3.34534
$766$	0	0
$767$	−26.3304	−0.950735
$768$	0	0
$769$	−7.64570	−0.275711	−0.137856	−	0.990452i	$-0.544021\pi$
$769$	−7.64570	−0.275711	−0.137856	+	0.990452i	$0.544021\pi$
$770$	0	0
$771$	7.35604	0.264921
$772$	0	0
$773$	−12.7578	−0.458867	−0.229433	−	0.973324i	$-0.573687\pi$
$773$	−12.7578	−0.458867	−0.229433	+	0.973324i	$0.573687\pi$
$774$	0	0
$775$	71.8817	2.58207
$776$	0	0
$777$	50.6363	1.81657
$778$	0	0
$779$	2.37145	0.0849662
$780$	0	0
$781$	−11.0379	−0.394967
$782$	0	0
$783$	−143.301	−5.12115
$784$	0	0
$785$	−38.3028	−1.36709
$786$	0	0
$787$	−27.0124	−0.962889	−0.481445	−	0.876476i	$-0.659888\pi$
$787$	−27.0124	−0.962889	−0.481445	+	0.876476i	$0.659888\pi$
$788$	0	0
$789$	−27.2769	−0.971084
$790$	0	0
$791$	32.8740	1.16886
$792$	0	0
$793$	41.5033	1.47382
$794$	0	0
$795$	−29.1081	−1.03236
$796$	0	0
$797$	−17.2729	−0.611837	−0.305919	−	0.952058i	$-0.598964\pi$
$797$	−17.2729	−0.611837	−0.305919	+	0.952058i	$0.598964\pi$
$798$	0	0
$799$	−5.71685	−0.202248
$800$	0	0
$801$	47.2255	1.66863
$802$	0	0
$803$	−14.9944	−0.529141
$804$	0	0
$805$	47.9405	1.68968
$806$	0	0
$807$	101.899	3.58702
$808$	0	0
$809$	−38.7686	−1.36303	−0.681515	−	0.731804i	$-0.738679\pi$
$809$	−38.7686	−1.36303	−0.681515	+	0.731804i	$0.738679\pi$
$810$	0	0
$811$	−15.4408	−0.542199	−0.271100	−	0.962551i	$-0.587387\pi$
$811$	−15.4408	−0.542199	−0.271100	+	0.962551i	$0.587387\pi$
$812$	0	0
$813$	−78.3725	−2.74864
$814$	0	0
$815$	31.6616	1.10906
$816$	0	0
$817$	−1.18026	−0.0412921
$818$	0	0
$819$	−98.9025	−3.45593
$820$	0	0
$821$	34.9424	1.21950	0.609750	−	0.792594i	$-0.291270\pi$
$821$	34.9424	1.21950	0.609750	+	0.792594i	$0.291270\pi$
$822$	0	0
$823$	20.9332	0.729684	0.364842	−	0.931069i	$-0.381123\pi$
$823$	20.9332	0.729684	0.364842	+	0.931069i	$0.381123\pi$
$824$	0	0
$825$	−24.5761	−0.855630
$826$	0	0
$827$	−31.8329	−1.10694	−0.553470	−	0.832869i	$-0.686697\pi$
$827$	−31.8329	−1.10694	−0.553470	+	0.832869i	$0.686697\pi$
$828$	0	0
$829$	−15.1128	−0.524891	−0.262445	−	0.964947i	$-0.584529\pi$
$829$	−15.1128	−0.524891	−0.262445	+	0.964947i	$0.584529\pi$
$830$	0	0
$831$	5.46529	0.189589
$832$	0	0
$833$	−5.07428	−0.175813
$834$	0	0
$835$	49.9085	1.72715
$836$	0	0
$837$	−149.479	−5.16676
$838$	0	0
$839$	−39.0512	−1.34820	−0.674099	−	0.738641i	$-0.735468\pi$
$839$	−39.0512	−1.34820	−0.674099	+	0.738641i	$0.735468\pi$
$840$	0	0
$841$	55.5412	1.91521
$842$	0	0
$843$	−43.6139	−1.50214
$844$	0	0
$845$	58.6935	2.01912
$846$	0	0
$847$	2.34459	0.0805611
$848$	0	0
$849$	43.4036	1.48961
$850$	0	0
$851$	−38.0983	−1.30599
$852$	0	0
$853$	−14.8573	−0.508703	−0.254351	−	0.967112i	$-0.581862\pi$
$853$	−14.8573	−0.508703	−0.254351	+	0.967112i	$0.581862\pi$
$854$	0	0
$855$	27.4045	0.937216
$856$	0	0
$857$	42.3513	1.44669	0.723347	−	0.690485i	$-0.242603\pi$
$857$	42.3513	1.44669	0.723347	+	0.690485i	$0.242603\pi$
$858$	0	0
$859$	−23.9371	−0.816723	−0.408361	−	0.912820i	$-0.633900\pi$
$859$	−23.9371	−0.816723	−0.408361	+	0.912820i	$0.633900\pi$
$860$	0	0
$861$	−18.2324	−0.621358
$862$	0	0
$863$	41.0021	1.39573	0.697865	−	0.716230i	$-0.254134\pi$
$863$	41.0021	1.39573	0.697865	+	0.716230i	$0.254134\pi$
$864$	0	0
$865$	9.06357	0.308171
$866$	0	0
$867$	18.3640	0.623676
$868$	0	0
$869$	5.18195	0.175786
$870$	0	0
$871$	12.0866	0.409540
$872$	0	0
$873$	−47.8116	−1.61818
$874$	0	0
$875$	20.6747	0.698933
$876$	0	0
$877$	−1.31855	−0.0445242	−0.0222621	−	0.999752i	$-0.507087\pi$
$877$	−1.31855	−0.0445242	−0.0222621	+	0.999752i	$0.507087\pi$
$878$	0	0
$879$	26.4992	0.893797
$880$	0	0
$881$	−25.5816	−0.861865	−0.430933	−	0.902384i	$-0.641815\pi$
$881$	−25.5816	−0.861865	−0.430933	+	0.902384i	$0.641815\pi$
$882$	0	0
$883$	23.2089	0.781040	0.390520	−	0.920594i	$-0.372295\pi$
$883$	23.2089	0.781040	0.390520	+	0.920594i	$0.372295\pi$
$884$	0	0
$885$	−56.0920	−1.88551
$886$	0	0
$887$	33.6865	1.13108	0.565540	−	0.824721i	$-0.308668\pi$
$887$	33.6865	1.13108	0.565540	+	0.824721i	$0.308668\pi$
$888$	0	0
$889$	13.0064	0.436222
$890$	0	0
$891$	27.8479	0.932941
$892$	0	0
$893$	−1.69320	−0.0566609
$894$	0	0
$895$	−1.61179	−0.0538760
$896$	0	0
$897$	103.208	3.44602
$898$	0	0
$899$	88.1863	2.94118
$900$	0	0
$901$	8.47887	0.282472
$902$	0	0
$903$	9.07417	0.301969
$904$	0	0
$905$	−29.7559	−0.989119
$906$	0	0
$907$	−40.5536	−1.34656	−0.673280	−	0.739388i	$-0.735115\pi$
$907$	−40.5536	−1.34656	−0.673280	+	0.739388i	$0.735115\pi$
$908$	0	0
$909$	53.2240	1.76533
$910$	0	0
$911$	14.4247	0.477911	0.238956	−	0.971031i	$-0.423195\pi$
$911$	14.4247	0.477911	0.238956	+	0.971031i	$0.423195\pi$
$912$	0	0
$913$	4.24982	0.140648
$914$	0	0
$915$	88.4150	2.92291
$916$	0	0
$917$	27.3549	0.903337
$918$	0	0
$919$	−39.7419	−1.31096	−0.655482	−	0.755211i	$-0.727534\pi$
$919$	−39.7419	−1.31096	−0.655482	+	0.755211i	$0.727534\pi$
$920$	0	0
$921$	−13.5340	−0.445962
$922$	0	0
$923$	60.0573	1.97681
$924$	0	0
$925$	−49.3611	−1.62298
$926$	0	0
$927$	−99.3574	−3.26332
$928$	0	0
$929$	−20.4506	−0.670964	−0.335482	−	0.942047i	$-0.608899\pi$
$929$	−20.4506	−0.670964	−0.335482	+	0.942047i	$0.608899\pi$
$930$	0	0
$931$	−1.50289	−0.0492552
$932$	0	0
$933$	35.1991	1.15237
$934$	0	0
$935$	11.9347	0.390305
$936$	0	0
$937$	−8.32869	−0.272086	−0.136043	−	0.990703i	$-0.543439\pi$
$937$	−8.32869	−0.272086	−0.136043	+	0.990703i	$0.543439\pi$
$938$	0	0
$939$	−23.6798	−0.772760
$940$	0	0
$941$	51.0299	1.66353	0.831763	−	0.555131i	$-0.187332\pi$
$941$	51.0299	1.66353	0.831763	+	0.555131i	$0.187332\pi$
$942$	0	0
$943$	13.7179	0.446716
$944$	0	0
$945$	−129.165	−4.20172
$946$	0	0
$947$	22.5652	0.733270	0.366635	−	0.930365i	$-0.380510\pi$
$947$	22.5652	0.733270	0.366635	+	0.930365i	$0.380510\pi$
$948$	0	0
$949$	81.5848	2.64835
$950$	0	0
$951$	−58.2380	−1.88850
$952$	0	0
$953$	34.1812	1.10724	0.553619	−	0.832770i	$-0.313246\pi$
$953$	34.1812	1.10724	0.553619	+	0.832770i	$0.313246\pi$
$954$	0	0
$955$	−63.5178	−2.05539
$956$	0	0
$957$	−30.1506	−0.974630
$958$	0	0
$959$	−6.63045	−0.214108
$960$	0	0
$961$	60.9886	1.96738
$962$	0	0
$963$	−27.0633	−0.872103
$964$	0	0
$965$	60.9762	1.96289
$966$	0	0
$967$	−22.0673	−0.709636	−0.354818	−	0.934935i	$-0.615457\pi$
$967$	−22.0673	−0.709636	−0.354818	+	0.934935i	$0.615457\pi$
$968$	0	0
$969$	−11.0716	−0.355670
$970$	0	0
$971$	−13.2403	−0.424902	−0.212451	−	0.977172i	$-0.568145\pi$
$971$	−13.2403	−0.424902	−0.212451	+	0.977172i	$0.568145\pi$
$972$	0	0
$973$	−36.3396	−1.16499
$974$	0	0
$975$	133.719	4.28243
$976$	0	0
$977$	6.65904	0.213041	0.106521	−	0.994310i	$-0.466029\pi$
$977$	6.65904	0.213041	0.106521	+	0.994310i	$0.466029\pi$
$978$	0	0
$979$	6.09139	0.194682
$980$	0	0
$981$	154.982	4.94820
$982$	0	0
$983$	−30.9907	−0.988449	−0.494225	−	0.869334i	$-0.664548\pi$
$983$	−30.9907	−0.988449	−0.494225	+	0.869334i	$0.664548\pi$
$984$	0	0
$985$	−24.3319	−0.775280
$986$	0	0
$987$	13.0178	0.414361
$988$	0	0
$989$	−6.82733	−0.217097
$990$	0	0
$991$	−59.2229	−1.88128	−0.940639	−	0.339408i	$-0.889773\pi$
$991$	−59.2229	−1.88128	−0.940639	+	0.339408i	$0.889773\pi$
$992$	0	0
$993$	19.2340	0.610373
$994$	0	0
$995$	53.2093	1.68685
$996$	0	0
$997$	−28.5785	−0.905092	−0.452546	−	0.891741i	$-0.649484\pi$
$997$	−28.5785	−0.905092	−0.452546	+	0.891741i	$0.649484\pi$
$998$	0	0
$999$	102.647	3.24761

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3344.2.a.w.1.1		6
4.3	odd	2		836.2.a.e.1.6	✓	6
12.11	even	2		7524.2.a.q.1.1		6
44.43	even	2		9196.2.a.n.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
836.2.a.e.1.6	✓	6	4.3	odd	2
3344.2.a.w.1.1		6	1.1	even	1	trivial
7524.2.a.q.1.1		6	12.11	even	2
9196.2.a.n.1.6		6	44.43	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3344.a (trivial)

\(f(q)\)	\(=\)	\(q-3.27915 q^{3} +3.53478 q^{5} +2.34459 q^{7} +7.75283 q^{9} +O(q^{10})\)	\(q-3.27915 q^{3} +3.53478 q^{5} +2.34459 q^{7} +7.75283 q^{9} +1.00000 q^{11} -5.44101 q^{13} -11.5911 q^{15} +3.37635 q^{17} +1.00000 q^{19} -7.68827 q^{21} +5.78459 q^{23} +7.49465 q^{25} -15.5853 q^{27} +9.19463 q^{29} +9.59107 q^{31} -3.27915 q^{33} +8.28761 q^{35} -6.58617 q^{37} +17.8419 q^{39} +2.37145 q^{41} -1.18026 q^{43} +27.4045 q^{45} -1.69320 q^{47} -1.50289 q^{49} -11.0716 q^{51} +2.51125 q^{53} +3.53478 q^{55} -3.27915 q^{57} +4.83925 q^{59} -7.62786 q^{61} +18.1772 q^{63} -19.2328 q^{65} -2.22139 q^{67} -18.9686 q^{69} -11.0379 q^{71} -14.9944 q^{73} -24.5761 q^{75} +2.34459 q^{77} +5.18195 q^{79} +27.8479 q^{81} +4.24982 q^{83} +11.9347 q^{85} -30.1506 q^{87} +6.09139 q^{89} -12.7569 q^{91} -31.4506 q^{93} +3.53478 q^{95} -6.16698 q^{97} +7.75283 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9}+O(q^{10}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9	\( 6 q - q^{3} + 5 q^{5} + 2 q^{7} + 17 q^{9} + 6 q^{11} + 8 q^{13} - 9 q^{15} - 2 q^{17} + 6 q^{19} + 18 q^{21} - 5 q^{23} + 13 q^{25} - 7 q^{27} + 10 q^{29} - 3 q^{31} - q^{33} + 4 q^{35} + 7 q^{37} + 10 q^{39} + 6 q^{41} - 16 q^{43} + 42 q^{45} + 12 q^{49} - 8 q^{51} + 20 q^{53} + 5 q^{55} - q^{57} - 15 q^{59} + 24 q^{61} + 20 q^{63} - 28 q^{65} - 25 q^{67} - 33 q^{69} + 9 q^{71} - 26 q^{73} - 28 q^{75} + 2 q^{77} + 16 q^{79} + 58 q^{81} + 2 q^{83} + 12 q^{85} + 36 q^{87} + 7 q^{89} - 8 q^{91} - 55 q^{93} + 5 q^{95} + 37 q^{97} + 17 q^{99}+O(q^{100}) \) 6 * q - q^3 + 5 * q^5 + 2 * q^7 + 17 * q^9 + 6 * q^11 + 8 * q^13 - 9 * q^15 - 2 * q^17 + 6 * q^19 + 18 * q^21 - 5 * q^23 + 13 * q^25 - 7 * q^27 + 10 * q^29 - 3 * q^31 - q^33 + 4 * q^35 + 7 * q^37 + 10 * q^39 + 6 * q^41 - 16 * q^43 + 42 * q^45 + 12 * q^49 - 8 * q^51 + 20 * q^53 + 5 * q^55 - q^57 - 15 * q^59 + 24 * q^61 + 20 * q^63 - 28 * q^65 - 25 * q^67 - 33 * q^69 + 9 * q^71 - 26 * q^73 - 28 * q^75 + 2 * q^77 + 16 * q^79 + 58 * q^81 + 2 * q^83 + 12 * q^85 + 36 * q^87 + 7 * q^89 - 8 * q^91 - 55 * q^93 + 5 * q^95 + 37 * q^97 + 17 * q^99

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